Introduction to Quadratic Functions

Q: What happens if $a = 0$?

Then there is no $x^{2}$ term, and $y = bx + c$ is just a linear function. That is exactly why the definition requires $a \neq 0$.

Q: Does a quadratic need all three terms?

No. Only the $x^{2}$ term is required. $y = x^{2}$, $y = 3x^{2} - 12$, and $y = -2x^{2} + 7x$ are all quadratic functions.

Q: What are the zeros of a quadratic?

The zeros are the $x$-values where $y = 0$ — the points where the parabola crosses the $x$-axis. A quadratic can have two zeros, one zero, or no real zeros.

Q: Does every parabola cross the $x$-axis?

No. A parabola that opens up with its vertex above the $x$-axis never crosses it, and the same goes for one that opens down with its vertex below. Those quadratics have no real zeros.

A quadratic function is any function that can be written as $y = ax^{2} + bx + c$ with $a \neq 0$ . The $x^{2}$ term is the giveaway — if the highest power of $x$ is exactly $2$ , the function is quadratic. Its graph is a U-shaped curve called a parabola.

Quadratics show up anywhere a quantity rises and then falls, or falls and then rises: the height of a thrown ball, the area of a garden as you change one side, profit as a price changes. This lesson is about recognizing a quadratic when you see one and reading the key features of its graph.

What makes a function quadratic

Check the powers of $x$ . In $y = ax^{2} + bx + c$ , the $a$ must not be zero — that $x^{2}$ term is required. The $bx$ term and the constant $c$ are optional, so $y = x^{2}$ , $y = 2x^{2} - 8$ , and $y = -x^{2} + 3x$ are all quadratic.

Compare that to the functions it gets confused with. $y = 3x + 5$ is linear — no squared term. $y = 2^{x}$ is exponential — the $x$ is in the exponent, which is completely different from $x$ being squared. $y = \dfrac{1}{x}$ has $x$ in a denominator, so it is not quadratic either.

Reading a parabola

The sign of $a$ tells you which way the parabola opens: if $a > 0$ it opens up, and if $a < 0$ it opens down.

The vertex is the turning point — the lowest point of a parabola that opens up, or the highest point of one that opens down. The $y$ -intercept is where the graph crosses the $y$ -axis, and it is always the constant $c$ , because plugging in $x = 0$ leaves only $c$ . The $x$ -intercepts, where the graph crosses the $x$ -axis, are called the zeros of the function.

The graph below is $y = x^{2} - 4x + 1$ . It opens up, its vertex is at $(2, -3)$ , it crosses the $y$ -axis at $1$ , and it crosses the $x$ -axis twice — so this quadratic has two zeros.

Worked examples

Example 1: is it quadratic?

Is $y = 5x - x^{2}$ a quadratic function?

Rewrite with the squared term first

y = -x^{2} + 5x

Identify

a

a = -1

Since

a \neq 0

, the function is quadratic

Answer: Yes — it is quadratic with $a = -1$ , $b = 5$ , $c = 0$ .

Example 2: direction of opening

Does $y = -3x^{2} + 5$ open upward or downward?

Identify

a

a = -3

The sign of

a

is negative

A negative

a

means the parabola opens downward

Answer: Downward.

Example 3: read the features

For $y = x^{2} - 4x + 1$ , give the direction, the $y$ -intercept, and the vertex (use the graph above).

Identify

a

— positive, so it opens up

a = 1

The

y

-intercept is the constant term

c = 1

Read the turning point from the graph

(2, -3)

Answer: Opens up, $y$ -intercept $1$ , vertex $(2, -3)$ .

Try one yourself

Common questions

What happens if

a = 0

Then there is no $x^{2}$ term, and $y = bx + c$ is just a linear function. That is exactly why the definition requires $a \neq 0$ .

Does a quadratic need all three terms?

No. Only the $x^{2}$ term is required. $y = x^{2}$ , $y = 3x^{2} - 12$ , and $y = -2x^{2} + 7x$ are all quadratic functions.

What are the zeros of a quadratic?

The zeros are the $x$ -values where $y = 0$ — the points where the parabola crosses the $x$ -axis. A quadratic can have two zeros, one zero, or no real zeros.

Does every parabola cross the

x

-axis?

No. A parabola that opens up with its vertex above the $x$ -axis never crosses it, and the same goes for one that opens down with its vertex below. Those quadratics have no real zeros.

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Introduction to Quadratic Functions

What makes a function quadratic

Reading a parabola

Worked examples

Example 1: is it quadratic?

Example 2: direction of opening

Example 3: read the features

Try one yourself

Common questions

Keep learning

Translations of Quadratic Functions

Dilations of Quadratic Functions

Vertex Form

Solve by Graphing